# The Evolution Postulate > How does a quantum state change when you *don't* measure it? Postulate 3 says: by a **unitary** > transformation. In continuous time this is the **Schrödinger equation**; in the discrete language of > quantum computing it is "apply a gate." The two are the same statement — $U(t) = e^{-iHt}$ — and the > link is precisely the "Hermitian generates unitary" theorem from Term 0. This postulate is why every > quantum gate is unitary and why (un-measured) quantum computation is reversible. ## Learning Objectives After this lesson you will be able to: 1. State **Postulate 3** in both discrete ($\lvert\psi'\rangle = U\lvert\psi\rangle$) and continuous (Schrödinger) forms. 2. Explain why evolution must be **unitary** (preservation of normalization/probability). 3. Solve the time-independent Schrödinger equation to obtain the propagator $U(t) = e^{-iHt}$. 4. Contrast reversible **unitary** evolution with irreversible **measurement**. 5. Evolve states numerically and confirm unitarity. --- ## Intuition Between measurements, a closed quantum system glides along deterministically — but the "glide" is a *rotation* of the state vector inside Hilbert space, never a stretch or shrink (that would change total probability). Rotations of complex vectors that preserve lengths and angles are exactly **unitary** maps. The generator of this continuous rotation is the system's energy operator, the **Hamiltonian** $H$. In quantum computing we usually skip the continuous story and just specify the net rotation — a **gate** $U$ — but physically that gate is "let the right Hamiltonian act for the right amount of time" (Course 1.6, and hardware in Term 4.4). --- ## Theory ### Postulate 3 (Unitary evolution) > **Postulate 3 (discrete form).** The evolution of a closed quantum system from time $t_1$ to $t_2$ > is given by a **unitary** operator $U$ depending only on $t_1,t_2$: > $$ \lvert\psi(t_2)\rangle = U\,\lvert\psi(t_1)\rangle, \qquad U^\dagger U = I. $$ > > **Postulate 3 (continuous form).** The time evolution of the state of a closed system is governed by > the **Schrödinger equation** > $$ \boxed{\, i\,\frac{d}{dt}\lvert\psi(t)\rangle = H\,\lvert\psi(t)\rangle \,} \qquad(\hbar = 1), $$ > where $H = H^\dagger$ is a fixed Hermitian operator, the **Hamiltonian** (the energy observable). We set $\hbar = 1$ throughout (see [Appendix C](../../appendix/C-notation-glossary.md)); restoring units puts $\hbar$ on the left: $i\hbar\,d_t\lvert\psi\rangle = H\lvert\psi\rangle$. ### Why unitary? Probability conservation Unitarity is forced by Postulate 1: a state must stay normalized. If $\lvert\psi'\rangle = U\lvert\psi\rangle$, then $\langle\psi'|\psi'\rangle = \langle\psi|U^\dagger U|\psi\rangle$, which equals $\langle\psi|\psi\rangle = 1$ for **all** states iff $U^\dagger U = I$. Equivalently ([0.1.6](../../term-0-foundations/c01-linear-algebra/06-special-operators.md)), unitaries are precisely the inner-product-preserving maps — they preserve all overlaps, hence all Born probabilities. Any non-unitary "evolution" would leak or create probability. ### From the Schrödinger equation to the propagator For a **time-independent** $H$, the Schrödinger equation $i\,d_t\lvert\psi\rangle = H\lvert\psi\rangle$ is a linear ODE with constant operator coefficient. Its solution is the operator exponential $$ \boxed{\ \lvert\psi(t)\rangle = e^{-iHt}\,\lvert\psi(0)\rangle \equiv U(t)\,\lvert\psi(0)\rangle\ }. $$ **Verification.** Differentiate the series $U(t) = e^{-iHt} = \sum_{n\ge0}\frac{(-iHt)^n}{n!}$ term by term: $\frac{d}{dt}U(t) = -iH\,e^{-iHt} = -iH\,U(t)$, so $i\,d_t(U\lvert\psi(0)\rangle) = H\,U\lvert\psi(0)\rangle$, and $U(0) = I$ gives the right initial condition. ∎ And $U(t)$ is unitary because $H$ is Hermitian — the exact theorem proved in [0.1.6](../../term-0-foundations/c01-linear-algebra/06-special-operators.md): $$ U(t)^\dagger U(t) = e^{+iH^\dagger t}e^{-iHt} = e^{+iHt}e^{-iHt} = I . $$ So the discrete and continuous forms agree: $U = e^{-iHt}$. Conversely, **Stone's theorem** (and finite dimensionally, just the matrix logarithm) says *every* unitary is $e^{-iH}$ for some Hermitian $H$ — so "a gate" and "evolve under some Hamiltonian for some time" are the same notion. This is the bridge from physics to circuits, fully exploited in Course 1.6 and Term 2. ### Composition and reversibility Unitaries compose: evolving by $U_1$ then $U_2$ is $U_2 U_1$, again unitary ([0.1.6 E2](../../term-0-foundations/c01-linear-algebra/06-special-operators.md)) — this is precisely a quantum **circuit**, a product of gates. And every unitary is **invertible**, with $U^{-1} = U^\dagger$: un-measured quantum evolution is **reversible**. Run the gates backwards (apply each $U^\dagger$ in reverse order) and you return to the start. Contrast **measurement** (Postulate 2), which collapses the state and destroys information — the one irreversible step in quantum theory. The interplay (reversible evolution + irreversible measurement) is what every algorithm orchestrates. > **Open systems caveat.** Postulate 3 is for *closed* systems. A system coupled to an environment > evolves non-unitarily (decoherence); that is described by quantum **channels** on density matrices — > Course 1.5 sets up the formalism and Term 4 develops noise. --- ## Worked Examples ### Example 1 — Evolution under $H = \tfrac{\omega}{2}Z$ With $H = \tfrac{\omega}{2}Z$, the propagator is (using the spectral form of $Z$, as in [0.1.6 Example 1](../../term-0-foundations/c01-linear-algebra/06-special-operators.md)): $$ U(t) = e^{-i\frac{\omega}{2}Z t} = \begin{pmatrix} e^{-i\omega t/2} & 0 \\ 0 & e^{+i\omega t/2}\end{pmatrix} = R_z(\omega t). $$ Apply to $\lvert+\rangle$: $U(t)\lvert+\rangle = \tfrac1{\sqrt2}(e^{-i\omega t/2}\lvert0\rangle + e^{+i\omega t/2}\lvert1\rangle)$. The *relative* phase $e^{i\omega t}$ between components grows linearly in time — the qubit's Bloch vector **precesses** about the $z$-axis at angular frequency $\omega$ (Course 1.6). Note $\langle Z\rangle$ stays $0$ throughout (the relative phase is invisible to a $Z$ measurement) but $\langle X\rangle = \cos\omega t$ oscillates — phase becomes observable in the $X$ basis. ### Example 2 — A gate as finite-time evolution To realize the **NOT** gate ($X$) we want $U = X$. Pick $H = \tfrac{\pi}{2}X$ and evolve for $t = 1$: $$ e^{-i\frac{\pi}{2}X} = \cos\tfrac{\pi}{2} I - i\sin\tfrac{\pi}{2} X = -iX, $$ which is $X$ up to the global phase $-i$ — physically the NOT gate. (The half-angle identity is from [Appendix E](../../appendix/E-math-identities.md).) This is the seed of **Rabi oscillations**: a resonant drive *is* a Hamiltonian, and letting it act for a calibrated time produces a gate ([1.6.2](../c06-dynamics/02-two-level-rabi.md)). --- ## Hands-on (Python) ```python import numpy as np from scipy.linalg import expm def dag(A): return A.conj().T def is_unitary(U): return np.allclose(dag(U) @ U, np.eye(U.shape[0])) Z = np.array([[1, 0], [0, -1]], dtype=complex) X = np.array([[0, 1], [1, 0]], dtype=complex) plus = np.array([1, 1], dtype=complex) / np.sqrt(2) # Propagator U(t) = exp(-iHt) for H = (ω/2) Z: omega = 2.0 def U(t, H): return expm(-1j * H * t) Ut = U(0.5, (omega / 2) * Z) print(is_unitary(Ut)) # True — evolution preserves probability psi_t = Ut @ plus print(np.round(np.abs(psi_t)**2, 4)) # [0.5 0.5] — populations unchanged under Rz... # ...but the relative phase shows up in : expX = lambda s: np.real(s.conj() @ X @ s) print(round(expX(psi_t), 4), "vs cos(ωt)=", round(np.cos(omega*0.5), 4)) # matches ``` ```python # A gate is finite-time evolution: exp(-i (π/2) X) = X up to global phase. gate = expm(-1j * (np.pi/2) * X) print(np.round(gate, 3)) # [[0,-i],[-i,0]] = -i X (physically NOT) print(np.allclose(np.abs(gate), np.abs(X))) # same operator up to phase # Reversibility: U† undoes U. ket0 = np.array([1, 0], dtype=complex) print(np.allclose(dag(Ut) @ (Ut @ ket0), ket0)) # True ``` > **On Braket.** Every gate you place in a `Circuit` is one of these unitaries; the SDK's > `Gate.RZ(angle)` is literally $R_z(\text{angle}) = e^{-i\,\text{angle}\,Z/2}$. Reversibility is why a > circuit can be *uncomputed* by appending the inverse gates — a trick used constantly from Term 2 on. --- ## Exercises **E1 (easy).** Verify that $U = e^{-i\theta X/2} = R_x(\theta)$ is unitary and compute $R_x(\pi)$ up to global phase.

Solution

$R_x(\theta) = \cos\tfrac\theta2 I - i\sin\tfrac\theta2 X$; $R_x^\dagger R_x = I$ since $X$ is Hermitian and $\theta$ real. $R_x(\pi) = -iX$, i.e. $X$ (NOT) up to the global phase $-i$.

**E2 (easy).** If $\lvert\psi(0)\rangle$ is an eigenstate of $H$ with energy $E$, find $\lvert\psi(t)\rangle$. What is observable about its time dependence?

Solution

$\lvert\psi(t)\rangle = e^{-iEt}\lvert\psi(0)\rangle$ — only a global phase accrues. Nothing observable changes: energy eigenstates are **stationary states**. (Relative phases between *different* energy eigenstates in a superposition *are* observable — Course 1.6.)

**E3 (medium).** Prove that the product of the propagators over $[0,t_1]$ and $[t_1, t_1+t_2]$ equals the propagator over $[0, t_1+t_2]$ for time-independent $H$.

Solution

$U(t_2)U(t_1) = e^{-iHt_2}e^{-iHt_1} = e^{-iH(t_1+t_2)} = U(t_1+t_2)$, valid because $H$ commutes with itself (so the exponents add — [Appendix E, BCH](../../appendix/E-math-identities.md)). ∎ This semigroup property is just "evolve, then evolve more."

**E4 (medium).** Show that if $H$ is *not* Hermitian (say $H = -i\Gamma$ with $\Gamma$ positive), $U(t) = e^{-iHt}$ is not unitary and the norm decays. Why does this violate Postulate 1?

Solution

$U(t) = e^{-i(-i\Gamma)t} = e^{-\Gamma t}$, with $U^\dagger U = e^{-2\Gamma t}\neq I$; norms shrink as $\lVert\psi(t)\rVert^2 = e^{-2\Gamma t}\lVert\psi(0)\rVert^2$. Total probability is no longer conserved, contradicting the requirement that states stay unit vectors. (Such non-Hermitian "effective Hamiltonians" are used as *models* of open systems, where the missing probability has leaked to an environment — Term 4.)

**E5 (hard).** Given a unitary $U$ on a qubit, show how to find a Hermitian $H$ with $U = e^{-iH}$, and apply it to $U = H_{\text{adamard}}$. (Use the spectral decomposition.)

Solution

Diagonalize $U = \sum_k e^{i\theta_k}\lvert v_k\rangle\langle v_k|$ (unitary ⇒ unit-modulus eigenvalues, [0.1.6](../../term-0-foundations/c01-linear-algebra/06-special-operators.md)). Define $H = \sum_k(-\theta_k)\lvert v_k\rangle\langle v_k|$ (real eigenvalues ⇒ Hermitian); then $e^{-iH} = \sum_k e^{i\theta_k}\lvert v_k\rangle\langle v_k| = U$. For the Hadamard, eigenvalues are $\pm1 = e^{i\cdot 0}, e^{i\pi}$ with the eigenvectors from [0.1.5 E4](../../term-0-foundations/c01-linear-algebra/05-eigen-spectral-theorem.md); so $H$ has eigenvalues $0$ and $-\pi$ on those eigenvectors, giving $H = \tfrac{\pi}{2}(I - \tfrac{1}{\sqrt2}(X+Z))$ up to an additive multiple of $I$ (which is just a global phase). ∎

--- ## Checkpoint 1. State Postulate 3 in discrete and continuous forms. 2. Why must closed-system evolution be unitary? 3. Solve $i\,d_t\lvert\psi\rangle = H\lvert\psi\rangle$ for time-independent $H$ and verify the solution. 4. In what sense are a "gate" and "evolution under a Hamiltonian" the same thing? 5. Contrast the reversibility of evolution with measurement.

Answers

1. Discrete: $\lvert\psi(t_2)\rangle = U\lvert\psi(t_1)\rangle$, $U$ unitary. Continuous: $i\,d_t\lvert\psi\rangle = H\lvert\psi\rangle$, $H$ Hermitian. 2. To preserve normalization/Born probabilities for all states, which requires $U^\dagger U = I$. 3. $\lvert\psi(t)\rangle = e^{-iHt}\lvert\psi(0)\rangle$; differentiating $e^{-iHt}$ gives $-iH e^{-iHt}$, satisfying the equation with $U(0)=I$. 4. $U(t) = e^{-iHt}$ for Hermitian $H$, and (Stone/matrix log) every unitary equals $e^{-iH}$ for some Hermitian $H$ — gates are net evolutions. 5. Unitary evolution is invertible ($U^{-1}=U^\dagger$), hence reversible; measurement collapses the state and is irreversible.

--- ## Further Reading - **[NC]** Nielsen & Chuang, §2.2.2 (Postulate 2, evolution) and §2.2.3. - **[Sak]** Sakurai & Napolitano, §2.1–2.2 — time evolution and the Schrödinger equation. - **[Gri]** Griffiths & Schroeter, Ch. 1–2 — the Schrödinger equation (with $\hbar$ explicit). --- ← Prev: [Observables & the Measurement Postulate](02-observables-measurement-postulate.md) · Up: [Term 1](../README.md) · Next: [The Composite Systems Postulate](04-composite-systems-postulate.md) →